Essays on threshold models with unit roots

Abstract

Empirical Time Series is too linear. After the 2008 great depression, the board members of the central banks realize that they were unable to foresee the fi nancial meltdown until it was too late, due to the linear structure of the models used for the forecasts, claiming the need for non-linear models. The fi eld of non-linear time series model is too vast, and sometimes these models are too complex to use them for forecasting. Furthermore, most of the economic variables are persistent, viewed as unit roots, adding an extra level of di culty in the study of non-linear time series models. The challenge is to develop non-linear models with persistent variables. Threshold models are a class of non-linear model characterized by different regimes, determined by a threshold variable. These regimes can represent the different stages of the economic cycles, for example, economic expansions and recessions, periods with high volatility and low volatility in the stock market, among many other examples. Many of the advantages of the threshold models are the simplicity of estimation using least square estimation, interpretation of the non-linear structure, and testing. In this dissertation, we study threshold models with unit roots from two different perspectives. In one had we introduce a univariate analysis and on the other hand, a multivariate analysis. In the fi rst chapter, titled "Threshold Stochastic Unit Roots Models" co-authored with Jesús Gonzalo and Raquel Montesinos, we present the univariate analysis by introducing a new class of stochastic unit-root (STUR) processes. This new model, namely the threshold autoregressive stochastic unit root (TARSUR) process, is strictly stationary, but if we do not consider the threshold effect, it can mislead to conclude that the process has a unit root. The TARSUR models are not only an alternative to xed unit root models but present interpretation, estimation, and testing advantages to the existent STUR models. This study analyzes the properties of the TARSUR models and proposes two simple tests to identify this type of processes. The fi rst test will allow us to detect the presence of unit roots, which can be fi xed or stochastic, and the asymptotic distribution (AD) of this test presents a distribution discontinuity depending if the unit root is fi xed or stochastic. The second test we propose is a simple t-statistic (or the supremum of a sequence of t-statistics) for testing the null hypothesis of a fi xed unit root versus a stochastic unit root hypothesis. It is shown that its asymptotic distribution (AD) depends if the threshold value is identi ed under the null hypothesis or not. When the threshold parameter is known, the AD is a standard normal distribution, while in the case of an unknown threshold value, the AD is a functional of Brownian Bridge. A Monte Carlo simulation shows that the proposed tests behave very well in a nite sample, and the Dickey-Fuller test cannot easily distinguish between exact unit roots and threshold stochastic unit roots. The study concludes with applications to U.S. stock prices, U.S. house prices, U.S. interest rates, and USD/Pound exchange rates. The second chapter, we present the multivariate analysis with "Multiple Long Run Equi- libria Through Cointegration Eyes". In this chapter, we introduce threshold effects in the cointegration relation. Cointegration has succeeded in capturing the unique long-run linear equilibrium. Speci c non-linearities have been incorporated into cointegrated models but always assuming the existence of a single equilibrium. In this study, we explore the possibility of different long-run equilibria depending on the state of the world (i.e., good and bad times, optimism and pessimism, frictional coordination) in a threshold framework. Starting from the present-value model (PVM) with different discount factors and depending on the state of the economy, we show that this type of PVM implies threshold cointegrated with different long-run equilibria. We present the estimation and inference theory, and the study nishes with two empirical applications where the variables are not linearly cointegrated but threshold cointegrated. The third chapter, we continue in the multivariate framework and introduce the paper titled "Quasi-Error Correction Model". Cointegration captures single long-run equilibrium relationships between economic variables and the error correction model (ECM) is the mechanism in which the equilibrium is maintained. In this study, we introduce the quasi-error correction model (QECM), derived from the cointegration relation with threshold effects, where each regime represents a different equilibrium relation between the variables. In contrast to the linear ECM, the QECM has a regressor which captures the switching between equilibria, capturing the dynamic structure of the equilibrium change. This regressor will pose a problem similar to the non-linear error correction models, where the model cannot be balanced using the traditional de nitions of integration. We present the estimation and the inference theory and nish with an empirical application for U.S. interest rate of instruments with different maturities.